My employer has a new logo (shown below). I do not have information on how this was created (as it was done by an outside company), though I'm fairly sure it was not done in any formal mathematical way:

It appears to be a triangular mesh of randomly spaced points, projected onto a sphere (at least to my eye, the points seem randomly distributed). I'd like to create something like this in Mathematica using built-in commands.

For random mesh, one could use randomly sampled points on a sphere and construct either DelaunayMesh or ConvexHullMesh from point set and use BoundaryMesh of that, but purely randomly sampled points don't actually produce aesthetic results. Thus, I use a truncated icosahedron data as an example.

It took a while from the first time I saw this question, but only because I only realized now that I had eventually built all the tools I thought I needed. Thus, this will link to a lot of my previous answers.

First, I'll use the Lloyd algorithm to generate a bunch of equidistributed points:

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